It is widely believed that maximum-likelihood estimators must be used to provide optimal estimates of power spectra. Since such estimators require the inversion and multiplication of Nd×Nd matrices, ...where Nd is the size of the data vector, maximum-likelihood estimators require at least of order N3d operations and become computationally prohibitive for Nd greater than a few tens of thousands. Because of this, a large and inhomogeneous literature exists on approximate methods of power-spectrum estimation. These range from manifestly suboptimal but computationally fast methods, to near-optimal but computationally expensive methods. Furthermore, much of this literature concentrates on the power-spectrum estimates rather than the equally important problem of deriving an accurate covariance matrix. In this paper, I consider the problem of estimating the power spectrum of cosmic microwave background (CMB) anisotropies from large data sets. Various analytic results on power-spectrum estimators are derived, or collated from the literature, and tested against numerical simulations. An unbiased hybrid estimator is proposed that combines a maximum-likelihood estimator at low multipoles and pseudo-Cℓ estimates at high multipoles. The hybrid estimator is computationally fast (i.e. it can be run on a laptop computer for Planck-sized data sets), nearly optimal over the full range of multipoles, and returns an accurate and nearly diagonal covariance matrix for realistic experimental configurations (provided certain conditions on the noise properties of the experiment are satisfied). It is argued that, in practice, computationally expensive methods that approximate the maximum-likelihood solution are unlikely to improve on the hybrid estimator, and may actually perform worse. The results presented here can be generalized to CMB polarization and to power-spectrum estimation using other types of data, such as galaxy clustering and weak gravitational lensing.
The amplitudes of the quadrupole and octopole measured from the Wilkinson Microwave Anisotropy Probe (WMAP) appear to be lower than expected according to the concordance Λ cold dark matter (ΛCDM) ...cosmology. However, the pseudo-Cℓ estimator used by the WMAP team is non-optimal. In this paper, we discuss the effects of Galactic cuts on pseudo-Cℓ and quadratic maximum likelihood estimators. An application of a quadratic maximum likelihood estimator to Galaxy-subtracted maps produced by the WMAP team and Tegmark, de Oliveira-Costa & Hamilton shows that the amplitudes of the low multipoles are stable to different Galactic cuts. In particular, the quadrupole and octopole amplitudes are found to lie in the ranges ΔT22= 176−250 (μK)2 and ΔT23= 794−1183 (μK)2 (and more likely to be at the upper ends of these ranges) rather than the values ΔT22= 123 (μK)2 and ΔT23= 611 (μK)2 found by the WMAP team. These results indicate that the discrepancy with the concordance ΛCDM model at low multipoles is not particularly significant and is in the region of a few per cent. This conclusion is consistent with an analysis of the low amplitude of the angular correlation function computed from quadratic maximum likelihood power spectrum estimates.
A model of supernova feedback during disc galaxy formation is developed. The model incorporates infall of cooling gas from a halo, and outflow of hot gas from a multiphase interstellar medium (ISM). ...The star formation rate is determined by balancing the energy dissipated in collisions between cold gas clouds with that supplied by supernovae in a disc marginally unstable to axisymmetric instabilities. Hot gas is created by thermal evaporation of cold gas clouds in supernova remnants, and criteria are derived to estimate the characteristic temperature and density of the hot component and hence the net mass outflow rate. A number of refinements of the model are investigated, including a simple model of a galactic fountain, the response of the cold component to the pressure of the hot gas, pressure-induced star formation and chemical evolution. The main conclusion of this paper is that low rates of star formation can expel a large fraction of the gas from a dwarf galaxy. For example, a galaxy with circular speed ∼50 km s−1 can expel ∼60–80 per cent of its gas over a time-scale of ∼1 Gyr, with a star formation rate that never exceeds ∼0.1 M⊙ yr−1. Effective feedback can therefore take place in a quiescent mode and does not require strong bursts of star formation. Even a large galaxy, such as the Milky Way, might have lost as much as 20 per cent of its mass in a supernova-driven wind. The models developed here suggest that dwarf galaxies at high redshifts will have low average star formation rates and may contain extended gaseous discs of largely unprocessed gas. Such extended gaseous discs might explain the numbers, metallicities and metallicity dispersions of damped Lyman α systems.
The temperature anisotropy power spectrum measured with the Wilkinson Microwave Anisotropy Probe (WMAP) at high multipoles is in spectacular agreement with an inflationary Λ-dominated cold dark ...matter cosmology. However, the low-order multipoles (especially the quadrupole) have lower amplitudes than expected from this cosmology, indicating a need for new physics. Here we speculate that the low quadrupole amplitude is associated with spatial curvature. We show that positively curved models are consistent with the WMAP data and that the quadrupole amplitude can be reproduced if the primordial spectrum truncates on scales comparable to the curvature scale.
There has been increasing interest by cosmologists in applying Bayesian techniques, such as Bayesian Evidence, for model selection. A typical example is in assessing whether observational data favour ...a cosmological constant over evolving dark energy. In this paper, the example of dark energy is used to illustrate limitations in the application of Bayesian Evidence associated with subjective judgements concerning the choice of model and priors. An analysis of recent cosmological data shows a statistically insignificant preference for a cosmological constant over simple dynamical models of dark energy. It is argued that for nested problems, as considered here, Bayesian parameter estimation can be more informative than computing Bayesian Evidence for poorly motivated physical models.
This paper generalizes the hybrid power spectrum estimation method developed in a previous paper to the estimation of polarization power spectra of the cosmic microwave background radiation. The ...hybrid power spectrum estimator is unbiased and we show that it is close to optimal at all multipoles, provided the pixel noise satisfies certain constraints. Furthermore, the hybrid estimator is computationally fast and can easily be incorporated in a Monte Carlo chain for Planck-sized data sets, which may be essential to account for realistic noise covariance and systematic errors. Simple formulae are given for the covariance matrices, including uncorrelated inhomogeneous instrumental noise, and these are tested extensively against numerical simulations. We compare the behaviour of simple pseudo-C
ℓ (PCL) estimates with maximum-likelihood estimates at low multipoles. For realistic sky cuts, maximum-likelihood estimates reduce very significantly the mixing of E and B modes. To achieve limits on the scalar-tensor ratio of r≪ 0.1 from sky maps with realistic sky cuts, maximum-likelihood methods, or PCL estimators based on unambiguous E and B modes, will be essential.
The magnitude—redshift relation for Type Ia supernovae is beginning to provide strong constraints on the cosmic densities contributed by matter, Ωm, and a cosmological constant, ΩΛ, although the ...results are highly degenerate in the Ωm—ΩΛ plane. Here we estimate the constraints that can be placed on a cosmological constant or quintessence-like component by extending supernovae samples to high redshift. Such measurements, when combined with constraints from anisotropies in the cosmic microwave background (CMB), could provide an important consistency check of systematic errors in the supernovae data. A large campaign of high-z supernovae observations with 10-m class telescopes could constrain Ωm to an accuracy (1σ) of 0.06 and ΩΛ to 0.15. A sample of supernovae at redshift z∼3, as might be achievable with a Next Generation Space Telescope, could constrain Ωm to an accuracy of about 0.02 independently of the value of ΩΛ. The constraints on a more general equation of state, wQpρ, converge slowly as the redshift of the supernovae data is increased. The most promising way of setting accurate constraints on wQ is by combining high-z supernovae and CMB measurements. With feasible measurements it should be possible to constrain wQ to a precision of about 0.06, if the Universe is assumed to be spatially flat. We use the recent supernovae sample of Perlmutter et al. and observations of the CMB anisotropies to constrain the equation of state in quintessence-like models via a likelihood analysis. The 2σ upper limits are wQ<−0.6 if the Universe is assumed to be spatially flat, and wQ<−0.4 for universes of arbitrary spatial curvature. The upper limit derived for a spatially flat Universe is close to the lower limit (wQ≈−0.7) allowed for simple potentials, implying that additional fine tuning may be required to construct a viable quintessence model.
The Wilkinson Microwave Anisotropy Probe (WMAP) has measured lower amplitudes for the temperature quadrupole and octopole anisotropies than expected in the best fitting (concordance) Λ-dominated cold ...dark matter (ΛCDM) cosmology. Some authors have argued that this discrepancy may require new physics. However, the statistical significance of this result is not clear. Some authors have applied frequentist arguments and claim that the discrepancy would occur by chance about 1 time in 700, if the concordance model is correct. Other authors have used Bayesian arguments to claim that the data show marginal evidence for new physics. I investigate these confusing and apparently conflicting claims in this Letter using a frequentist analysis and a simplified Bayesian analysis. On either analysis, I conclude that the WMAP results are consistent with the concordance ΛCDM model.
Operational high‐resolution numerical weather prediction models are now able to partially resolve turbulent motions due to increased computing power. The partitioning of resolved and parametrized ...fluxes becomes important in the representation of turbulent transfer that determines the state of the atmospheric boundary layer. In this study, successive simulations of a convective boundary layer using the Met Office Large Eddy Model from the large‐eddy simulation to the mesoscale limit are compared with the corresponding coarse‐grained fields to examine model behaviour over the grey zone. The differences in the turbulent kinetic energy partitioning between coarse‐grained (reference) and actual fields are identified and used to quantify sub‐grid diffusion in the grey zone under different atmospheric forcings and surface heat flux. It is shown that the excessive mixing due to the large values of mixing length at coarse resolutions results in the cut‐off of resolved turbulence in the grey zone. The damping of resolved motions comes earlier for wind shear runs. In contrast, coarse‐grained fields exhibit a smooth transition of the resolved turbulent kinetic energy across the scales. Decreasing numerical dissipation through the sub‐grid scheme leads to the increase of resolved turbulence, but fails to reproduce the reference transition pattern that imposes some physical limitations to the partially resolved turbulence simulations. Pragmatic blending of mixing length values maintains a more realistic turbulent kinetic energy transition from fine to coarse resolutions and potential temperature profiles in the grey zone. Bounding vertical diffusion to its effective values, an approach based on maintaining inherent properties of the flow across the scales, is able to match the coarse‐grained fields from the highly resolved to the almost unresolved state, regardless of the forcing. Finally, the complexity of modelling in the grey zone is exhibited in the dependence of turbulence onset on time and vertical resolution.
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